To recap, the argument is
1. Anything that has a beginning has a cause.
2. The universe has a beginning.
Therefore,
3. The universe has a cause.
In previous posts I have discussed the notions of validity and soundness. I have defended (1), and I have begun to defend (2). I leave it to the reader to go back and review these posts. I will continue my defense of (2).
If the universe did not have a beginning, then it must extend infinitely into the past. There are several reasons to think that this could not be the case. First, the required notion of infinity is implausible once applied to the physical world. To see this, we must distinguish between two competing concepts of infinity, and we need set theory to do that.
Set theory is the logic of sets, classes, or groups of things. Sets can range over anything. For example, there is the set of even numbers, the set of people that attend Forest Hills Community Church, the set of dogs, etc. Some sets are large, e.g., the set of all humans. Some sets are empty, e.g., the set of unicorns. Sets are entirely defined in terms of their members. The set of dogs is defined in terms of all dogs. And, since the set of unicorns and the set of griffins have exactly the same members (both sets are empty), there is really only one set: the empty set. To get at the distinction between types of infinity, we will be concerned primarily with sets of numbers.
The first type of infinity is a “potential infinity.” Potential infinites are what we are most familiar with. Think about starting at 1 and counting to the very last number. You can’t because there is no last number. You could count forever. Nevertheless, your counting has a definite beginning, the number 1, and regardless of how high you count, you’ll have only counted a finite range of numbers (e.g., from 1 to 10, 1 to 143,532, etc.). The key is to remember that potentially infinite series are always finite but could be extended indefinitely into the future.
Potential infinites are not actual infinites.
Actual infinites are another matter entirely, and we’ve got to go back to set theory.
Let’s define the sets A and B and C as:
A = {1, 2, 5}
B = {43, 3, 8}
C = {2, 5}
First, note that A, B, and C do not have exactly the same members, so they are not identical sets. However, A and B have the same number of terms. A and B have three members each. Since they share the same number of terms, A and B can be put in one-to-one correspondence. Each term from each set can be paired together.
1/43, 2/3, 5/8
However, B and C cannot be put into one-to-one correspondence because there are more terms in B than there are in C.
43/2, 3/5, 8/-.
Beyond the notion of one-to-one correspondence, we need to understand the notion of “proper subset.” Take some set X and some set Y. X is a proper subset of Y if Y shares all of X’s members, but X does not share all of Y’s members.
By this definition, we can see that C is a proper subset of A. A shares all of C’s members (2 and 5), but C does not share all of A’s members (C does not include 1).
We are now ready for the definition of an actual infinite. An actually infinite set is one that can be put into one-to-one correspondence with the set of Natural Numbers (N) and with a proper subset of itself. For example, the set of even numbers (E) is an actually infinite set.
Take the set of Natural Numbers, the set of even numbers, and a proper subset of the even numbers (F):
N = {0, 1, 2, 3, 4, 5, ...}
E = {2, 4, 6, 8, 10, 12, ...}
F = {4, 6, 8, 10, 12, 14, ...}
N, E, and F all share the same number of terms so they can all be put in one-to-one correspondence with each other. See also that F is a proper subset of E. E and F differ only with respect to one member of E, the number 2. Since every member of F is a member of E, and one member of E is not a member of F, then, by definition, F is a proper subset of E. Therefore, since E can be put into one-to-one correspondence with both N and F, E is an actually infinite set.
Why all the tedious theory? I need to draw out an important point. There is a HUGE difference between potentially infinite series and actually infinite sets. Remember, sets are defined by their members, and that means that they have all of their members all at once. Potentially infinite series can always be extended, but the actual number of terms in the series at any one time will always constitute a finite set. Potential infinites are not infinite sets at all.
If the universe had a first moment, then there would be a finite number of moments between that first moment and the present moment. But if the universe extends backward into the infinite past, then it has no first moment. But this means that there must be an infinite number of moments that have actually transpired up to this very moment. And that means that we have to use the notion of the actually infinite set and not that of the potentially infinite series when dealing with a universe without a beginning.
But does this even make sense? No it does not!
You get all kinds of crazy things happening once you admit actually infinite sets being real in the physical world. I’ll give just one of a number of possible examples.
Imagine a library with an actually infinite number of books. Half of the books are red; half are black. Now, how many black books are there? Well, there are an infinite number of black books. How about the number of red books? Same thing right? Sure. But surely, when you add the number of black books with the number of red books you get a bigger number, right? No, the total number is identical to the number of books in each half: infinity.
Is your head spinning yet?
Now, if we have an infinite number of books in the library, and somebody checks a book in, how many books are there now? That’s right! There’s still an infinite number. What about if someone checks a book out? Still infinite.
Sigh...
If you don’t get the example, that’s good. You’re not supposed to. The point is that physically realized actually infinites don’t make any sense in the real world. If it won’t work for books, it won’t work the universe. But if we can’t use the notion of actually infinite sets, then we’re back to using potentially infinite series. But potentially infinite series have first members. So the universe would have to have a first member. That means the universe would have to have a beginning.
There is more to be said, but it’ll have to wait until next time.
I’m certainly available for discussion. These are complex issues, and I want you to understand. Leave comments, send emails, etc.
